Test of scaling exponents for percolation-cluster perimeters

Abstract

The formula for the fractal dimensionality of the perimeter (hull) of a percolation cluster, ${\mathrm{d}}_{\mathrm{f}}^{\mathcal{’}}$=1+1/ν, proposed recently by Sapoval, Rosso, and Gouyet, is shown to imply for the perimeter scaling exponents τ’=1+2ν/(1+ν), σ’=1/(1+ν), and γ’=2. Monte Carlo simulations of very large perimeter-generating walks yield τ’=2.143±0.002 and ${\mathrm{d}}_{\mathrm{f}}^{\mathcal{’}}$=1.751±0.002, consistent with these predictions (on the assumption that ν=(4/3)). The walks are also used to determine ${\mathrm{p}}_{\mathrm{c}}$=0.5927 5±0.0000 3 for site percolation on a square lattice.